failure function tests

1 FAILURE FUNCTION

Let ϕ⁢(x) be a function of x. Then, x=ψ⁢(x0) is a failure function if the values of x generated by ψ⁢(x0), when substituted in ϕ⁢(x), generate only failures in accordance with our definition of a failure. Here x0 is a specific value of x.

1.1 Examples

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(i) Let the mother function be a polynomial
in x (coeffficients belong to 𝒵 ), say ϕ⁢(x).
Let our definition of a failure be a composite number. Then, x=ψ⁢(x0)=x0+k⁢(ϕ⁢(x0)) is a failure function because the values of x generated by ϕ⁢(x0), when substituted in ϕ⁢(x) , generate only failures.

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(ii) Let the mother function be an exponential function, say ϕ⁢(x)=ax+c.
Then x=ψ⁢(x0)=x0+k.E⁢u⁢l⁢e⁢r⁢p⁢h⁢i⁢(ϕ⁢(x0)) is a failure function since the values of x generated by ψ⁢(x0), when substituted in the mother function, generate only failures.

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(iii) Let our definition of a failure be a non-Carmichael number.
Let the mother function ϕ⁢(x) be 2n+49. Then, n=5+6⁢k is its failure function ψ⁢(x).

1.2 Note

Here too our definition of a failure is a composite number and k belongs to N.